Let $X$ be a scheme which is regular in codimension 1 ( it is integral and seperated as well). In other words, for every point in $x \in X$, the local ring at $x$, $\mathcal O_{X,x}$ is a dvr. Now, let $Y$ be a closed integral subscheme of $X$ and let $y$ be its generic point.
Let $\eta$ be the generic point of $X$. Let $O_{X,\eta}$ be $K(X)$ be the field of rational functions of $X$. Why is that the fraction field of $\mathcal O_{X,y}$ is same as K(X)? I do understand there is an injection from $\mathcal O_{X,y}$ is same as K(X), but I'm not sure how is $K(X)$ exactly the fraction field of $\mathcal O_{X,y}$?
Let $A $ be an integral domain. Then, $(0) $ is a prime ideal and the fraction field of $A $ is the same as $A $ localized at $(0) $. In this case the generic point corresponds to the $(0) $ ideal.